Problem: $ B = \left[\begin{array}{rr}4 & 3 \\ 2 & 1\end{array}\right]$ $ F = \left[\begin{array}{rr}4 & 5 \\ -2 & -1\end{array}\right]$ What is $ B F$ ?
Answer: Because $ B$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ B F = \left[\begin{array}{rr}{4} & {3} \\ {2} & {1}\end{array}\right] \left[\begin{array}{rr}{4} & \color{#DF0030}{5} \\ {-2} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{4}+{3}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{4}+{3}\cdot{-2} & ? \\ {2}\cdot{4}+{1}\cdot{-2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{4}+{3}\cdot{-2} & {4}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{-1} \\ {2}\cdot{4}+{1}\cdot{-2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{4}+{3}\cdot{-2} & {4}\cdot\color{#DF0030}{5}+{3}\cdot\color{#DF0030}{-1} \\ {2}\cdot{4}+{1}\cdot{-2} & {2}\cdot\color{#DF0030}{5}+{1}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}10 & 17 \\ 6 & 9\end{array}\right] $